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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Denote the steady state solution as <span class="process-math">\(U(x)\text{,}\)</span> then it satisfies the following two-point boundary value problem</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
U''=0,\quad U(0)=2,\quad U(1)=4,
\end{equation*}
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<p class="continuation">which gives the solution</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
U(x) = 2 + 2x.
\end{equation*}
</div>
<p class="continuation">Let <span class="process-math">\(w(x,t)\)</span> be the solution of the heat equation with homogeneous boundary condition. Then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
w_t=w_{xx},\quad w(0,t)=w(1,t)=0,\quad w(x,0)=u(x,0)-U(x)=-\sin\pi x-3\sin 3\pi x.
\end{equation*}
</div>
<p class="continuation">The formal solution for <span class="process-math">\(w\)</span> is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
w(x,t)=\sum_{n=1}^{\infty}C_n e^{-n^2\pi^2 t}\sin(n\pi x),\qquad n=1,2,3,\cdots
\end{equation*}
</div>
<p class="continuation">Here <span class="process-math">\(C_n\)</span> are Fourier coefficients of the initial data <span class="process-math">\(w(x,0)\text{.}\)</span> We find only two coefficients that are not 0, namely</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
C_1=-1,\quad C_3=-3.
\end{equation*}
</div>
<p class="continuation">i.e.</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
w(x,t)=-e^{-\pi^2 t}\sin(\pi x) - 3e^{-9\pi^2 t}\sin (3\pi x).
\end{equation*}
</div>
<p class="continuation">Finally combine <span class="process-math">\(w(x,t)\)</span> together with <span class="process-math">\(U(x)\text{,}\)</span> we get the solution</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,t)=w(x,t)+U(x)=2+2x-e^{-\pi^2 t}\sin(\pi x) - 3e^{-9\pi^2 t}\sin (3\pi x).
\end{equation*}
</div>
<span class="incontext"><a href="sec7_8.html#p-425" class="internal">in-context</a></span>
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